有限差分法求解声子热输运方程
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摘要
声子辐射输运方程是一个微分-积分方程,使用有限差分法数值求解具有边界条件的声子辐射输运方程,可借助Gauss-Seidel方法和有限差分离散化来保证得到稳定的数值解.通过对室温下Ge/Si/Ge薄膜的声子热输运过程进行一维数值模拟,以及对界面处使用漫射失配界面模型,可以得到温度沿着薄膜法向方向的变化曲线和Ge/Si薄膜厚度比例与温度在界面处跳跃值对材料整体结构的热导率的影响,以及随着薄膜厚度的增加薄膜热导率的变化趋势;对硅薄膜声子热输运过程进行二维数值模拟,可以得到温度沿着薄膜法向和面向方向的分布情况,以及当薄膜宽度和厚度比例不同时温度的变化和法向热导率的变化.
The Equation of Phonon Radiative Transfer is a differential-integral equation.The finite difference method is used to numerically solve the Equation of Phonon Radiative Transfer with boundary conditions.The Gauss-Seidel method and the finite difference discretization can be used to ensure stable numerical solutions.Through one-dimensional numerical simulation of the phonon heat transport process of Ge/Si/Ge film at room temperature and by using the diffuse mismatch interface model at the interface,it can evaluate the influence of the temperature along the normal direction of the film and the thickness ratio of the Ge/Si film and the temperature jump at the interface on the thermal conductivity of the overal-1 structure of the material,and the change of the thermal conductivity of the film as the thickness of the film increases.The two-dimensional numerical simulation of the phonon heat transport process of the silicon film can obtain the temperature along the the normal and orientation-oriented distribution of the film,and the change in temperature and the thermal conductivity when the film width and thickness ratio are different.
The Equation of Phonon Radiative Transfer is a differential-integral equation.The finite difference method is used to numerically solve the Equation of Phonon Radiative Transfer with boundary conditions.The Gauss-Seidel method and the finite difference discretization can be used to ensure stable numerical solutions.Through one-dimensional numerical simulation of the phonon heat transport process of Ge/Si/Ge film at room temperature and by using the diffuse mismatch interface model at the interface,it can evaluate the influence of the temperature along the normal direction of the film and the thickness ratio of the Ge/Si film and the temperature jump at the interface on the thermal conductivity of the overal-1 structure of the material,and the change of the thermal conductivity of the film as the thickness of the film increases.The two-dimensional numerical simulation of the phonon heat transport process of the silicon film can obtain the temperature along the the normal and orientation-oriented distribution of the film,and the change in temperature and the thermal conductivity when the film width and thickness ratio are different.
引文
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[2]韩亚芬.纳米复合隔热材料导热机理与特性研究[D].哈尔滨工业大学,2013:13-37.
[3]Venkatasubramanian R.Chapter 4--Phonon Blocking Electron Transmitting Superlattice Structures as Advanced Thin Film Thermoelectric Materials[J].Semiconductors&Semimetals,2001,71:175-201.
[4]Chen G.Nanoscale energy transport and conversion[M].Oxford University Press,2005:348-366.
[5]Miyazaki K,Arashi T,Makino D,et al.Heat conduction in micro-structured materials[J].IEEE Transactions on Components&Packaging Technologies,2006,29(2):247-253.
[6]Yilbas B S,Mansoor S B. Phonon Transport in Two-Dimensional Silicon-Diamond Film Pair[J].Journal of Thermophysics&Heat Transfer,2013,27(3):465-473.
[7]Haider Ali.Heat transport characteristics in thin films[D].King Fand University of Petroleum&Minerals,2014:74-199.
[8]Chu A J H.Phonon transport in two-dimensional silicon thin film:influence of film width and boundary conditions on temperature distribution[J].European Physical Journal B,2012,85(7):1-9.
[9]R.Yang,et al.Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation[J].Journal of Heat Transfer,2005,127(3):298-306.
[10]裴志坚.基于高斯-赛德尔迭代法及MATLAB软件的电路方程组求解方法[J].北京工业职业技术学院学报,2017,16(3):22-25.
[11]吴专保.线性方程组的几种数值方法的MATLAB程序[J].荆楚理工学院学报,2007,22(3):78-81.
[12]杨洋.界面热阻的晶格动力学模拟[D].东南大学,2012:6-36.
[13]张兴丽,孙兆伟.基于分子动力学对超晶格结构界面热阻的模拟研究[J].航空材料学报,2011,31(4):7-10.
[14]赵兰萍,徐烈,李兆慈,等.固体界面间接触导热的机理和应用研究[J].低温工程,2000,(4):29-34.
[15]M.Mohamadi,et al.Numerical Study of Thermal Conductivity Reduction in Nanolayered Si-Ge Structures by Solving the Equation of Phonon Radiative Transfer[J].Journal of the Serbian Society for Computational Mechanics,2017,11(1):29-45.
[16]冉鑫,郭洋裕,王沫然.双层薄膜界面声子输运的Monte Carlo模拟[J].工程热物理学报,2017,V38(4):841-846.
[17]Nield D A,Bejan A.Heat Transfer Through a Porous Medium[M].Convection in Porous Media.Springer New York,2017,31-46.